Lie Groupoids in Classical Field Theory I: Noether's Theorem

Lie groupoids in classical field theory I: Noether’s theorem

“…The construction of the "curvature map" for connections in a given fiber bundle E over M is similar but somewhat more complicated because it involves its semiholonomous second order jet bundle J 2 E. To see how that goes, we proceed as in Ref. [6] by first constructing the iterated jet bundle J(JE) of E and noting that this allows two projections to JE, namely, the iterated jet target projection π J(JE) : J(JE) −→ JE as well as the jet prolongation Jπ JE : J(JE) −→ JE of the jet target projection π JE : JE −→ E : then by definition, J 2 E is the subset of J(JE) where these two projections coincide. Concretely, for e ∈ E, u e ∈ J e E and u ′ ue ∈ J ue (JE),…”

“…equations (89) and (90) of Ref. [6].) Combining this with the natural action of the linear frame groupoid GL(T M ) of the base manifold M on the cotangent bundle T * M of M , we obtain an induced action of the Lie groupoid GL(T M ) × M G on the linearized jet bundle JE of E,…”

“…equations (96) and (98) of Ref. [6].) On the other hand, applying the jet functor to all structural maps that appear in the original action (10), we obtain an induced action…”

“…In the first paper of this series [6], we have initiated an investigation of how to handle symmetries -or more precisely, local symmetries -in classical field theories using the language of Lie groupoids and their actions. However, the formalism developed there is perhaps a bit too general because it allows us to leave the nature of the underlying Lie groupoids and their actions completely unspecified, whereas there can be no doubt that the motivation for the entire program comes predominantly from one single (class of) example(s), namely, gauge theories.…”

“…Next, we introduce the (first order) jet groupoid JG of G and use the results of the previous section and of Ref. [6] to write down natural actions of JG on various derived bundles such as the jet bundle JP and the connection bundle CP of P or the jet bundle JE of any bundle E associated to P . We also show how iterating this procedure provides induced actions of the second order jet groupoid J 2 G and, more generally, the semiholonomous second order jet groupoid J 2 G of G on the semiholonomous second order jet bundle J 2 P and on the (first order) jet bundle J(CP ) of the connection bundle CP of P .…”